‘There’s no “there” there!’
When in doubt, plot! The plot of a funciton tells essentially everything that needs to be known about its behaviour.
The first and third functions never reach +1 so there is no solution to either of them.
%variables
syms X Y;
solve (cos(X+0.5)+Y==0.8,X)
solve (sin(Y)-2*X==1.6,Y)
%convergence
G1(Y)=-acos(4/5 - Y) - 1/2;
G2(X)=pi - asin(2*X + 8/5);
%Derivative
G1X=diff(G1,X)
G1Y=diff(G1,Y)
G2X=diff(G2,X)
G2Y=diff(G2,Y)
%convergence limits
eq1 = -2/(1 - (2*X + 8/5)^2)^(1/2)-1
figure
fplot(eq1,[-2 0])
grid
ylim([-25, 5])
solve(-2/(1 - (2*X + 8/5)^2)^(1/2)==1, X)
solve(-2/(1 - (2*X + 8/5)^2)^(1/2)==-1)
eq2 = -1/(1 - (Y - 4/5)^2)^(1/2)-1
solve(-1/(1 - (Y - 4/5)^2)^(1/2)==1)
figure
fplot(eq2,[-0.5 2])
grid
ylim([-25, 5])
solve(-1/(1 - (Y - 4/5)^2)^(1/2)==-1)
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